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Theorem conncompss 22807
Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
conncompss ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem conncompss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp1 1137 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑋)
2 conntop 22791 . . . . . . 7 ((𝐽t 𝑇) ∈ Conn → (𝐽t 𝑇) ∈ Top)
323ad2ant3 1136 . . . . . 6 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝐽t 𝑇) ∈ Top)
4 restrcl 22531 . . . . . . 7 ((𝐽t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
54simprd 497 . . . . . 6 ((𝐽t 𝑇) ∈ Top → 𝑇 ∈ V)
6 elpwg 4567 . . . . . 6 (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
73, 5, 63syl 18 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
81, 7mpbird 257 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 ∈ 𝒫 𝑋)
9 3simpc 1151 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn))
10 eleq2 2823 . . . . . 6 (𝑦 = 𝑇 → (𝐴𝑦𝐴𝑇))
11 oveq2 7369 . . . . . . 7 (𝑦 = 𝑇 → (𝐽t 𝑦) = (𝐽t 𝑇))
1211eleq1d 2819 . . . . . 6 (𝑦 = 𝑇 → ((𝐽t 𝑦) ∈ Conn ↔ (𝐽t 𝑇) ∈ Conn))
1310, 12anbi12d 632 . . . . 5 (𝑦 = 𝑇 → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) ↔ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn)))
14 eleq2 2823 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
15 oveq2 7369 . . . . . . . 8 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
1615eleq1d 2819 . . . . . . 7 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
1714, 16anbi12d 632 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1817cbvrabv 3416 . . . . 5 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)}
1913, 18elrab2 3652 . . . 4 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn)))
208, 9, 19sylanbrc 584 . . 3 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
21 elssuni 4902 . . 3 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
2220, 21syl 17 . 2 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
23 conncomp.2 . 2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
2422, 23sseqtrrdi 3999 1 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Conn) → 𝑇𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  {crab 3406  Vcvv 3447  wss 3914  𝒫 cpw 4564   cuni 4869  (class class class)co 7361  t crest 17310  Topctop 22265  Conncconn 22785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-rest 17312  df-top 22266  df-conn 22786
This theorem is referenced by:  conncompcld  22808  tgpconncompeqg  23486  tgpconncomp  23487
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